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Abstract
We have developed and implemented a (J, B) space Newton method to solve the full nonlinear three dimensional magnetohydrodynamic equilibrium equations in toroidal geometry. Various cases have been run successfully, demonstrating significant improvement over Picard iteration, including a 3D stellarator equilibrium at β = 2%. The algorithm first solves the equilibrium force balance equation for the current density J, given a guess for the magnetic field B. This step is taken from the Picard-iterative PIES 3D equilibrium code. Next, we apply Newton's method to Ampere's Law by expansion of the functional J(B), which is defined by the first step. An analytic calculation in magnetic coordinates, of how the Pfirsch-Schlüter currents vary in the plasma in response to a small change in the magnetic field, yields the Newton gradient term (analogous to ∇f · δx in Newton's method for f(x) = 0). The algorithm is computationally feasible because we do this analytically, and because the gradient term is flux surface local when expressed in terms of a vector potential in an [special characters omitted] gauge. The equations are discretized by a hybrid spectral/offset grid finite difference technique, and leading order radial dependence is factored from Fourier coefficients to improve finite-difference accuracy near the polar-like origin. After calculating the Newton gradient term we transfer the equation from the magnetic grid to a fixed background grid, which greatly improves the code's performance.